Lower bounds on the smallest eigenvalue of a sample covariance
        matrix.

Yaskov, Pavel

doi:10.1214/ecp.v19-3807

Cited by 32 publications

(21 citation statements)

References 7 publications

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“…with probability ≥ 1 − 1 n , where K = K(p, B) > 0 depends only on p, B. Note that better estimates for the smallest singular value were previously obtained in [17] and later strengthened in [31,32]. The papers [26] and [17] were apparently the first ones where lower bounds for the smallest singular value were given in quite a general setting without any restrictions on the magnitude of the matrix norm A N 2→2 .…”

Section: Corollarymentioning

confidence: 97%

Sample Covariance Matrices of Heavy-Tailed Distributions

Tikhomirov

2017

International Mathematics Research Notices

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Section: Corollarymentioning

confidence: 97%

Sample Covariance Matrices of Heavy-Tailed Distributions

Tikhomirov

2017

International Mathematics Research Notices

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“…For non-asymptotic results in this direction, we refer the reader to papers [11,19] for the case of i.i.d. entries (see also [28] where no moment conditions are assumed); [1,2] for log-concave distributions of rows and [21,12,8,30,5] for more general isotropic distributions. We refer to surveys [18,29] (see also [17]) for more information.…”

Section: Introductionmentioning

confidence: 99%

Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Rebrova

Tikhomirov

2018

Isr. J. Math.

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Let A = (a ij ) be an n × n random matrix with i.i.d. entries such that Ea 11 = 0 and Ea 11 2 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B n 2 of cardinality at most exp(δn) such that with probability very close to one we haveIn fact, a stronger statement holds true. As an application, we show that for some L ′ > 0 and u ∈ [0, 1) depending only on the distribution law of a 11 , the smallest singular value s n of the matrix A satisfies P{s n (A) ≤ εn −1/2 } ≤ L ′ ε + u n for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

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“…One expects weaker assumption for estimating the smallest singular value. This already appeared in the work [31] and was pushed further in recent works [19,33,35] and in [14,20,26] which led to new bounds on the performance of ℓ 1 -minimization methods.…”

Section: Introduction and Main Resultsmentioning

confidence: 95%

On the interval of fluctuation of the singular values of random matrices

Guédon¹,

Litvak²,

Pajor³

et al. 2017

J. Eur. Math. Soc.

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Let A be a matrix whose columns X 1 , . . . , X N are independent random vectors in R n . Assume that the tails of the 1-dimensional marginals decay as P(| X i , a | ≥ t) ≤ t −p uniformly in a ∈ S n−1 and i ≤ N . Then for p > 4 we prove that with high probability A/ √ n has the Restricted Isometry Property (RIP) provided that Euclidean norms |X i | are concentrated around √ n. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio n/N . Moreover, we obtain sharp bounds for both problems when the decay is of the type exp(−t α ) with α ∈ (0, 2], extending the known case α ∈ [1, 2].

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Lower bounds on the smallest eigenvalue of a sample covariance matrix.

Abstract: We provide tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components.

Cited by 32 publications

References 7 publications

Sample Covariance Matrices of Heavy-Tailed Distributions

Sample Covariance Matrices of Heavy-Tailed Distributions

Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

On the interval of fluctuation of the singular values of random matrices

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